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In algebraic geometry, a
motive (or sometimes motif,
following French usage) denotes refers to 'some
essential part of an algebraic variety'. To date, pure
motives have been defined, while conjectural mixed motives have
not. Pure motives are triples (X, p, m), where X
is a smooth projective variety, p : X ⊢
X is an idempotent correspondence, and m an
integer. A morphism from (X, p, m) to (Y, q, n)
is given by a correspondence of degree n - m.

As far as mixed motives, mathematicians are working to find a
suitable definition which will then provide a "universal" cohomology theory. In terms of category
theory, it was intended to have a definition via splitting idempotents in a category of
algebraic correspondences. The way
ahead for that definition has been blocked for some decades, by the
failure to prove the standard
conjectures on algebraic cycles. This prevents the category
from having 'enough' morphisms, as can currently be shown. While
the category of motives was supposed to be the universal Weil cohomology much discussed in
the years 1960-1970, that hope for it remains unfulfilled. On the
other hand, by a quite different route, motivic
cohomology now has a technically-adequate definition.

can be put on increasingly solid mathematical footing with a
deep meaning. Of course, the above equations are already known to
be true in many senses, such as in the sense of CW-complex where "+" corresponds to
attaching cells, and in the sense of various cohomology theories,
where "+" corresponds to the direct sum.

From another viewpoint, motives continue the sequence of
generalizations from rational functions on varieties to divisors on
varieties to Chow groups of varieties. The generalization happens
in more than one direction, since motives can be considered with
respect to more types of equivalence than rational equivalence. The
admissiable equivalences are given by the definition of an adequate equivalence
relation.

Definition of pure
motives

The category of pure motives often
proceeds in three steps. Below we describe the case of Chow motives
Chow(k), where k is any field.

First step: category of (degree 0) correspondences,
Corr(k)

The objects of Corr(k) are simply smooth projective
varieties over k. The morphisms are correspondences. They
generalize morphisms of varieties X → Y, which
can be associated with their graphs in X × Y, to
fixed dimensional Chow
cycles on X × Y.

It will be useful to describe correspondences of arbitrary
degree, although morphisms in Corr(k) are correspondences
of degree 0. In detail, let X and Y be smooth
projective varieties, let
be the decomposition of X into connected components, and
let di := dim Xi. If
r ∈ Z, then the correspondences of degree
r from X to Y are

.

Correspondences are often denoted using the "⊢"-notation, e.g.,
α : X ⊢ Y. For any α ∈
Corrr(X, Y) and β ∈ Corrs(Y,
Z), their composition is defined by

,

where the dot denotes the product in the Chow ring (i.e.,
intersection).

Returning to constructing the category Corr(k), notice
that the composition of degree 0 correspondences is degree 0. Hence
we define morphisms of Corr(k) to be degree 0
correspondences.

The association,

,

where Γf ⊆ X × Y is the graph of
f : X → Y, is a functor.

Just like SmProj(k), the category Corr(k) has
direct sums ()
and tensor
products (X ⊗ Y := X ×
Y). It is a preadditive category (see the convention for
preadditive vs. additive in the preadditive category article.) The
sum of morphisms is defined by

Third
step: category of pure Chow motives, Chow(k)

To proceed to motives, we adjoin to
Choweff(k) a formal inverse (with respect to
the tensor product) of a motive called the Lefschetz motive. The
effect is that motives become triples instead of pairs. The
Lefschetz motive L is

.

If we define the motive 1, called the
trivial Tate motive, by 1 :=
h(Spec(k)), then the pleasant equation

holds, since 1 ≅
(P1, P1 ×
pt). The tensor inverse of the Lefschetz motive is known
as the Tate motive, T :=
L-1. Then we define the category of pure Chow
motives by

Chow(k):
=
Chowef
f(k)[T].

A motive is then a triple (X ∈ SmProj(k),
p : X ⊢ X, n ∈
Z) such that p ˆ p = p. Morphisms are
given by correspondences

,

and the composition of morphisms comes from composition of
correspondences.

Other
types of motives

In order to define an intersection product, cycles must be
"movable" so we can intersect them in general position. Choosing an
suitable equivalence relation on cycles will guarantee that every
pair of cycles has an equivalent pair in general position that we
can intersect. The Chow groups are defined using rational
equivalence, but other equivalences are possible, and each defines
a different sort of motive. Examples of equivalences, from
strongest to weakest, are

Rational equivalence

Algebraic equivalence

Smash-nilpotence equivalence (sometimes called Voevodsky
equivalence)

Homological equivalence (in the sense of Weil cohomology)

Numerical equivalence

The literature occasionally calls every type of pure motive a
Chow motive, in which case a motive with respect to algebraic
equivalence would be called a Chow motive modulo algebraic
equivalence.

Mixed
motives

For a fixed base field k, the category of mixed
motives is a conjectural abelian tensor categoryMM(k),
together with a contravariant functor

Var(k) → MM(X)

taking values on all varieties (not just smooth projective ones
as it was the case with pure motives). This should be such that
motivic cohomology defined by

Ext∗MM(1, ?)

coincides with the one predicted by algebraic K-theory, and
contains the category of Chow motives in a suitable sense (and
other properties). The existence of such a category was conjectured
by Beilinson. This category is yet to
be constructed.

Instead of constructing such a category, it was proposed by Deligne to
first construct a category DM having the properties one
expects for the derived category

Db(MM(k)).

Getting MM back from DM would then be
accomplished by a (conjectural) motivic t-structure.

The current state of the theory is that we do have a suitable
category DM. Already this category is useful in
applications. Voevodsky'sFields medal-winning
proof of the Milnor conjecture uses these motives
as a key ingredient.

There are different definitions due to Hanamura, Levine and
Voevodsky. They are known to be equivalent in most cases and we
will give Voevodsky's definition below. The category contains Chow
motives as a full subcategory and gives the "right" motivic
cohomology. However, Voevodsky also shows that (with integral
coefficients) it does not admit a motivic t-structure.

Start with the category Sm of smooth varieties over a
perfect field. Similarly to the construction of pure motives above,
instead of usual morphisms smooth correspondences are
allowed. Compared to the (quite general) cycles used above, the
definition of these correspondences is more restrictive; in
particular they always intersect properly, so no moving of cycles
and hence no equivalence relation is needed to get a well-defined
composition of correspondences. This category is denoted
SmCor, it is additive.

As a technical intermediate step, take the bounded homotopy categoryKb(SmCor) of complexes of smooth schemes and
correspondences.

Apply localization of categories to force any variety
X to be isomorphic to X × A1 (product
with the affine line) and also, that a Mayer-Vietoris-sequence
holds, i.e. X = U ∪ V (union of two open
subvarieties) shall be isomorphic to U ∩ V →
U ⊔ V.

Finally, as above, take the pseudo-abelian envelope.

The resulting category is called the category of effective
geometric motives. Again, formally inverting the Tate object,
one gets the category DM of geometric
motives.

Explanation for
non-mathematicians

A commonly applied technique in mathematics is to study objects
carrying a particular structure by introducing a category whose morphisms
preserve this structure. Then one may ask, when are two given
objects isomorphic and ask for a "particularly nice" representative
in each isomorphism class. The classification of algebraic
varieties, i.e. application of this idea in the case of algebraic varieties, is very difficult due
to the highly non-linear structure of the objects. The relaxed
question of studying varieties up to birational isomorphism has led
to the field of birational geometry. Another way to
handle the question is to attach to a given variety X an
object of more linear nature, i.e. an object amenable to the
techniques of linear algebra, for example a vector space. This
"linearization" goes usually under the name of
cohomology.

There are several important cohomology theories which reflect
different structural aspects of varieties. The (partly conjectural)
theory of motives is an attempt to find a
universal way to linearize algebraic varieties, i.e. motives are
supposed to provide a cohomology theory which embodies all these
particular cohomologies. For example, the genus of a smooth projective curveC which is an interesting
invariant of the curve, is an integer, which can be read off the
dimension of the first Betti cohomology
group of C. So, the motive of the curve should contain the
genus information. Of course, the genus is a rather coarse
invariant, so the motive of C is more than just this
number.

The search for a
universal cohomology

Each algebraic variety X has a corresponding motive
[X], so the simplest examples of motives are:

The general idea is that one motive has the
same structure in any reasonable cohomology theory with good formal
properties; in particular, any Weil cohomology
theory will have such properties. There are different Weil
cohomology theories, they apply in different situations and have
values in different categories, and reflect different structural
aspects of the variety in question:

Betti cohomology is defined for varieties over (subfields of)
the complex
numbers, it has the advantage of being defined over the integers
and is a topological invariant

de Rham cohomology (for varieties over ℂ) comes with a mixed Hodge structure, it is a
differential-geometric invariant

All these cohomology theories share common properties, e.g.
existence of Mayer-Vietoris-sequences, homotopy
invariance
(H∗(X)≅H∗(X
× A1), the product of X with the affine
line) and others. Moreover, they are linked by comparison
isomorphisms, for example Betti cohomology
HBetti∗(X, ℤ/n) of
a smooth variety X over ℂ with finite coefficients is
isomorphic to l-adic cohomology with finite
coefficients.

The theory of motives is an attempt to find a
universal theory which embodies all these particular cohomologies
and their structures and provides a framework for "equations"
like

[projective line] = [line]+[point].

In particular, calculating the motive of any variety X
directly gives all the information about the several Weil
cohomology theories
HBetti∗(X),
HDR∗(X) etc.

Beginning with Grothendieck, people have tried to precisely
define this theory for many years.

where n and m are integers and ℤ(m)
is the m-th tensor power of the Tate object ℤ(1), which in
Voevodsky's setting is the complex ℙ1 → point
shifted by -2, and [n] means the usual shift in the triangulated
category.

Conjectures related to
motives

The standard
conjectures were first formulated in terms of the interplay of
algebraic cycles and Weil cohomology theories. The category of pure
motives provides a categorical framework for these conjectures.

The standard conjectures are commonly considered to be very hard
and are open in the general case. Grothendieck, with Bombieri,
showed the depth of the motivic approach by producing a conditional
(very short and elegant) proof of the Weil conjectures (which are proven by
different means by Deligne), assuming the standard conjectures
to hold.

Conjecture D, stating the concordance of numerical and
homological equivalence, implies the
equivalence of pure motives with respect to homological and
numerical equivalence. (In particular the former category of
motives would not depend on the choice of the Weil cohomology
theory). Jannsen (1992) proved the following unconditional result:
the category of (pure) motives over a field is abelian and
semisimple if and only if the chosen equivalence relation is
numerical equivalence.

The Hodge
conjecture, may be neatly reformulated using motives: it holds
iff the
Hodge realization mapping any pure motive with rational
coefficients (over a subfield k of ℂ) to its Hodge
structure is a full functorH :
M(k)ℚ → HSℚ (rational Hodge
structures). Here pure motive means pure motive with respect to
homological equivalence.

Similarly, the Tate conjecture is equivalent to: the
so-called Tate realization, i.e. ℓ-adic cohomology is a faithful
functor H : M(k)ℚℓ →
Repℓ(Gal(k)) (pure motives up to homological
equivalence, continuous representations of the absolute Galois group of the
base field k), which takes values in semi-simple
representations. (The latter part is automatic in the case of the
Hodge analogue).

Tannakian
formalism and motivic Galois group

To motivate the (conjectural) motivic Galois group, fix a field
k and consider the functor

finite separable extensions K of k → finite sets
with a (continuous) action of the absolute Galois group of
k

which maps K to the (finite) set of embeddings of
K into an algebraic closure of k. In Galois theory this
functor is shown to be an equivalence of categories. Notice that
fields are 0-dimensional. Motives of this kind are called
Artin motives. By ℚ-linearizing the above objects, another
way of expressing the above is to say that Artin motives are
equivalent to finite ℚ-vector spaces together with an action of the
Galois group.

The objective of the motivic Galois group is to
extend the above equivalence to higher-dimensional varieties. In
order to do this, the technical machinery of Tannakian
category theory (going back to Tannaka-Krein
duality, but a purely algebraic theory) is used. Its purpose is
to shed light on both the Hodge conjecture and the Tate
conjecture, the outstanding questions in algebraic cycle
theory. Fix a Weil cohomology theory H. It gives a functor
from Mnum (pure motives using numerical
equivalence) to finite-dimensional ℚ-vector spaces. It can be shown
that the former category is a Tannakian category. Assuming the
equivalence of homological and numerical equivalence, i.e. the
above standard conjecture D, the functor H is an
exact faithful tensor-functor. Applying the Tannakian formalism,
one concludes that Mnum is equivalent to the
category of representations of an algebraic group
G, which is called motivic Galois group.

It is to the theory of motives what the Mumford-Tate group is to
Hodge theory.
Again speaking in rough terms, the Hodge and Tate conjectures are
types of invariant theory (the spaces that are
morally the algebraic cycles are picked out by invariance under a
group, if one sets up the correct definitions). The motivic Galois
group has the surrounding representation theory. (What it is not,
is a Galois group;
however in terms of the Tate conjecture and Galois representations on étale
cohomology, it predicts the image of the Galois group, or, more
accurately, its Lie
algebra.)